Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

In this paper we introduce and study a cohomology theory {Hn(−,A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A, n)}n≥0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−, n) is right adjoint to the functor ℘n, where ℘n(X•) is defined as the fundamental groupoid of the n-loop complex (X•). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with πi(Y ) = 0 for all i = n, n+ 1 and n ≥ 3; and also we obtain a classification theorem for those spaces: [−, Y ] ∼= Hn(−, ℘n(Y )). Mathematics Subject Classifications (2000): 18D10, 18G50, 18G60.